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Author Broyden, C. G. (Charles George)

Title Krylov solvers for linear algebraic systems / by Charles George Broyden, Maria Teresa Vespucci.

Imprint Amsterdam ; Boston : Elsevier, 2004.

Copies

Location Call No. OPAC Message Status
 Axe Elsevier ScienceDirect Ebook  Electronic Book    ---  Available
Edition 1st ed.
Description 1 online resource (330 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
Series Studies in computational mathematics ; 11
Studies in computational mathematics ; 11.
Bibliography Includes bibliographical references (pages 315-325) and index.
Note Print version record.
Contents Contents -- 1. Introduction. -- 2. The long recurrences. -- 3. The short recurrences. -- 4. The Krylov aspects. -- 5. Transpose-free methods. -- 6. More on QMR. -- 7. Look-ahead methods. -- 8. General block methods. -- 10. And in practice?? -- 11. Preconditioning. -- 12. Duality. -- Appendices. -- A. Reduction of upper Hessenberg matrix to upper triangular form by plane rotations. -- B. Schur complements. -- C. The Jordan form. -- D. Chebychev polynomials. -- E. The companion matrix. -- F. Algorithmic details.
Summary The first four chapters of this book give a comprehensive and unified theory of the Krylov methods. Many of these are shown to be particular examples of the block conjugate-gradient algorithm and it is this observation that permits the unification of the theory. The two major sub-classes of those methods, the Lanczos and the Hestenes-Stiefel, are developed in parallel as natural generalisations of the Orthodir (GCR) and Orthomin algorithms. These are themselves based on Arnoldi's algorithm and a generalised Gram-Schmidt algorithm and their properties, in particular their stability properties, are determined by the two matrices that define the block conjugate-gradient algorithm. These are the matrix of coefficients and the preconditioning matrix. In Chapter 5 the"transpose-free" algorithms based on the conjugate-gradient squared algorithm are presented while Chapter 6 examines the various ways in which the QMR technique has been exploited. Look-ahead methods and general block methods are dealt with in Chapters 7 and 8 while Chapter 9 is devoted to error analysis of two basic algorithms. In Chapter 10 the results of numerical testing of the more important algorithms in their basic forms (i.e. without look-ahead or preconditioning) are presented and these are related to the structure of the algorithms and the general theory. Graphs illustrating the performances of various algorithm/problem combinations are given via a CD-ROM. Chapter 11, by far the longest, gives a survey of preconditioning techniques. These range from the old idea of polynomial preconditioning via SOR and ILU preconditioning to methods like SpAI, AInv and the multigrid methods that were developed specifically for use with parallel computers. Chapter 12 is devoted to dual algorithms like Orthores and the reverse algorithms of Hegedus. Finally certain ancillary matters like reduction to Hessenberg form, Chebychev polynomials and the companion matrix are described in a series of appendices. comprehensive and unified approach up-to-date chapter on preconditioners complete theory of stability includes dual and reverse methods comparison of algorithms on CD-ROM objective assessment of algorithms
Subject Equations -- Numerical solutions.
Algebras, Linear.
Équations -- Solutions numériques.
Algèbre linéaire.
MATHEMATICS -- Algebra -- Elementary.
Algebras, Linear
Equations -- Numerical solutions
Added Author Vespucci, M. T. (Maria Teresa)
Other Form: Print version: Broyden, C.G. (Charles George). Krylov solvers for linear algebraic systems. 1st ed. Amsterdam ; Boston : Elsevier, 2004 0444514740 (OCoLC)56425871
ISBN 1423709330 (electronic bk.)
9781423709336 (electronic bk.)
9780444514745
0444514740
0080478875 (electronic bk.)
9780080478876 (electronic bk.)
1281008850
9781281008855
Standard No. AU@ 000048130929
AU@ 000050768030
AU@ 000051568438
AU@ 000069301920
CHNEW 001004798
DEBBG BV036962341
DEBBG BV039830209
DEBBG BV042317219
DEBSZ 276812565
DEBSZ 482349964
NZ1 12435122
NZ1 15192801

 
    
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